Error correction method for cross-shaped magnetic gradient tensor measurement system

Abstract

The invention relates to the technical field of geomagnetic vector measurement, and discloses an error correction method of a cross-shaped magnetic gradient tensor measurement system. According to the method, a coupling model containing a sensitivity error, a non-orthogonal error, a zero offset error and an array non-alignment error of a single fluxgate sensor is established, and a fitness function is designed to solve error parameters by utilizing the characteristics of high global search capability and high convergence speed of a firefly optimization algorithm (FA). Compared with the existing particle swarm optimization and the least square method, the method has the advantages that the correction precision of zero offset and non-alignment errors is remarkably improved, the single-sensor zero offset error estimation deviation is smaller than 2%, the array non-alignment angle error deviation is smaller than 1.5%, expensive equipment support is not needed, and real-time on-site implementation can be achieved. The method is suitable for the fields of magnetic anomaly detection, geophysical exploration and the like, and has the advantages of low cost, simplicity and convenience in operation and remarkable correction effect.

Description

Technical Field

[0001] The present invention relates to the technical field of geomagnetic vector measurement, and particularly to an error correction method for a cross-shaped magnetic gradient tensor measurement system. Background Art

[0002] In recent years, with the development of magnetic anomaly detection technology, the advantages of magnetic gradient tensor measurement have gradually emerged. Compared with traditional magnetic measurements (i.e., vector and total magnetic field strength measurements), magnetic gradient tensor measurement is relatively insensitive to azimuth errors and geomagnetic gradients, and the gradient tensor elements can provide valuable additional information, with better resolution. Based on the above advantages, magnetic gradient tensor measurement systems have been widely used in geophysical exploration, such as underwater target detection and the identification of unexploded ordnance.

[0003] The performance of a magnetic gradient tensor measurement system is adversely affected by measurement errors caused by specific factors. Therefore, before officially using the magnetic gradient tensor measurement system for positioning, it needs to be calibrated. First, the fluxgate sensor itself has non-orthogonal errors, sensitivity errors, and zero-offset errors; second, due to limitations in processing levels, when making brackets of different structures, it is not possible to ensure that the measurement axes of several sensors are exactly aligned with the orthogonal coordinate system of the measurement system, which causes misalignment errors between the sensors. The above errors will directly affect the measurement accuracy of the magnetic gradient tensor positioning system, so it must be calibrated before use.

[0004] A magnetic gradient tensor positioning system is always composed of multiple fluxgate sensors. Therefore, the error correction process can be divided into two steps: the first step is to correct a single fluxgate sensor, and the second step is to correct the misalignment errors of each magnetometer. A large number of literatures have focused on the calibration of magnetometer arrays, and the calibration methods are mainly divided into two categories: vector calibration methods and scalar calibration methods. The vector calibration method mainly provides a non-magnetic environment through a magnetic shielding barrel or provides a standard magnetic field through a Helmholtz coil, and combines a high-precision non-magnetic turntable to provide an accurate rotation angle for calibration. Therefore, the calibration equipment required by the vector calibration method is expensive, the calibration procedure is complex, which increases the cost required for sensor calibration and cannot achieve real-time on-site calibration. In contrast, the scalar calibration method perfectly avoids the problems of high cost and inconvenience for real-time on-site calibration. Because the scalar calibration method only requires a high-precision proton magnetometer or a high-precision magnetometer to monitor the background geomagnetic field, and has a simple calibration process and is easy to execute. For the calibration of a single magnetometer, scholars have successively proposed many calibration methods, such as the partitioned beetle antenna algorithm, the support vector regression method, the model coefficient automatic search method, and calibration based on the extended Kalman filter algorithm, etc.

[0005] In the scalar calibration method for misalignment errors of a magnetometer array, there are mainly two methods for solving error parameters. One is to directly design a fitness function using the magnetic gradient tensor matrix (hereinafter referred to as Method 1), and then use various methods to solve the error parameters; the other is to use a certain sensor as a reference, design a fitness function using the differences between the remaining sensors and the reference sensor, and then use different methods to solve the error parameters (hereinafter referred to as Method 2). In recent years, more scholars have used Method 1 for calibration. In 2020, Yanxia Liu proposed a two-step calibration method for magnetometers based on the particle swarm optimization algorithm. First, the fitness function of the particle swarm was designed using the amplitude invariance of the earth's magnetic field to obtain preliminary error parameters; second, the fitness function of the particle swarm was designed using the short-term invariance principle of the earth's magnetic field to obtain the rotation matrix, thereby uniquely determining the error matrix. This method can effectively calculate the error matrix, but it does not consider the problem that the particle swarm optimization algorithm is prone to falling into local optimal solutions. In 2023, Cheng Chi proposed using the total least squares method to correct the errors of a cross-shaped magnetic gradient tensor system. First, the least squares method was used to correct a single fluxgate sensor, and then the orthogonal Procrustes problem was used to calibrate the misalignment error; the linearized least squares method for correcting parameters is an algebraic method, and the effect of solving parameters is very good, but it ignores the correlation problem between various parameters; in 2023, Yu Huang proposed a two-step calibration method for a magnetic vector gradiometer based on the function-linked artificial neural network and the least squares method. First, the FLANN method was used to correct a single magnetometer, and then the least squares method was used to correct the misalignment error between sensors. This method can effectively correct a magnetic vector gradiometer composed of two fluxgate sensors, but it does not verify the calibration effect of this method for a cross-shaped magnetic vector gradiometer. However, the error parameters obtained by the fitness function used in Method 1 are only valid for the calculation of the magnetic gradient tensor matrix, and no real calibration is achieved. The research on calibration using Method 2 is relatively scarce. In 2013, Hongfeng Pang proposed a calibration process. First, the 'fsolve' function in MATLAB for solving nonlinear functions and the Levenberg-Marquard algorithm were used to solve the error parameters of a single fluxgate sensor, then the coordinate transformation between magnetometers was performed, and finally, with Sensor 1 as the reference sensor, the misalignment error angles of the remaining sensors were calculated by solving nonlinear equations in sequence; this method can effectively correct the errors of a magnetic gradient tensor positioning system, but the layout of its various sensors may introduce other unnecessary errors.

[0006] Therefore, the present invention proposes an error correction method for a cross-shaped magnetic gradient tensor measurement system based on the firefly optimization algorithm to solve the above technical problems. Summary of the Invention

[0007] The present invention provides an error correction method for a cross-shaped magnetic gradient tensor measurement system, aiming to solve the problem of insufficient measurement accuracy caused by multi-error coupling in the cross-shaped magnetic gradient tensor measurement system.

[0008] Since the measurement accuracy of the magnetic gradient tensor measurement system is affected by both the sensor's own error and the misalignment error of the magnetometer array, it is necessary to correct the magnetic gradient tensor measurement system before use. Currently, the calibration methods for the magnetic gradient tensor measurement array mainly include two categories: vector calibration method and scalar calibration method. The vector calibration method requires equipment such as a magnetic shielding barrel and a high-precision non-magnetic turntable, with high calibration costs, complex calibration processes, and unable to achieve real-time on-site calibration. Therefore, the present invention adopts a more economical and convenient scalar calibration method.

[0009] In the scalar calibration method, the selection of the optimization algorithm is also particularly important. Currently, researchers generally use the least squares method and the particle swarm optimization algorithm. However, the least squares method has the disadvantages of being sensitive to the initial value and slow convergence speed, and the particle swarm optimization algorithm is prone to falling into a local optimal solution during operation. Therefore, the present invention uses a new optimization algorithm, the firefly optimization algorithm, to avoid the above defects.

[0010] The present invention achieves the above object through the following technical solutions.

[0011] The error correction method for the cross-shaped magnetic gradient tensor measurement system provided by the present invention is as follows:

[0012] 1. Establish an error correction model

[0013] For the self-error of the fluxgate sensor, considering the triaxial non-orthogonal error, sensitivity error, and zero bias error comprehensively (as Figure 1 shown, it is the non-orthogonal error model diagram of the triaxial fluxgate sensor), the relationship between the measured value and the true value of the sensor is obtained as follows:

[0014] B m = KAB i + b + ε (1)

[0015] In formula (1), let B m be the measured value of the fluxgate sensor, B i be the actual magnetic field value, represents the sensitivity parameter matrix, b = [b x , b y , b z T is the zero bias error, is the angle conversion matrix used to describe the non-orthogonal error, and ε is the measurement noise of the fluxgate sensor itself.

[0016] ​By simplifying Equation (1), the error correction model of the fluxgate sensor can be derived:

[0017] B m = TB i + b + ε (2)

[0018] In Equation (2), T is the comprehensive error matrix. Since the measurement noise of the fluxgate sensor is very small compared to its inherent error and can usually be ignored, the error model can be further simplified to:

[0019] B i = T -1 (B m - b) (3)

[0020] Equation (3) represents the final error correction model of the fluxgate sensor. It can be seen from this that only by calculating the inverse matrix T -1 and the zero-bias vector b, the measured value of the sensor can be converted into the true value, thus completing the error correction of a single fluxgate sensor.

[0021] For the misalignment error of the cross-shaped magnetometer array, as Figure 2 (a) shows, the cross-shaped magnetometer array is designed by four three-axis magnetic sensors. The error caused by the inconsistent characteristics of the sensors due to the configuration structure (offset and deflection of the installation center point) during the arrangement and installation of different sensitive axes of each sensor is called the misalignment error. Among them, the displacement error can be eliminated by high-precision manufacturing processes, while the rotational misalignment error is difficult to avoid.

[0022] The orthogonal transformation relationship diagram can be represented by Figure 2 (b). The output orientation of the three-axis sensor is not exactly aligned with the orthogonal coordinate system of the magnetic gradient tensor system. However, the two orthogonal systems in any attitude in space can be transformed by a rotation involving three misalignment angles. Let the standard coordinate system be O-X0Y0Z0. First, rotate by an angle ψ around the X-axis to get O-X1Y1Z1, then rotate by an angle around the Y-axis to get O-X2Y2Z2, and finally rotate by an angle θ around the Z-axis to get the coordinate system to be corrected O-X3Y3Z3. Defining the rotation around the X-axis as the roll transformation, the rotation around the Y-axis as the pitch transformation, and the rotation around the Z-axis as the azimuth transformation.

[0023] Among them, ψ is the roll angle, is the pitch angle, and θ is the azimuth angle. Assuming that there is only misalignment error between the sensors in the cross-shaped array, the transformation relationship between the two orthogonal coordinate systems is as follows:

[0024]

[0025] Among them, R1 is the roll rotation matrix, R2 is the pitch rotation matrix, and R3 is the azimuth rotation matrix. With the above conversion relationships, the magnetic fields in two non-aligned orthogonal coordinate systems can be converted according to them. If the rotation sequence is first the x-axis, then the y-axis, and finally the z-axis, the conversion relationship can be expressed as:

[0026]

[0027] Among them, B r represents the measured value with error, and B i represents the ideal value. From the above formula, we can get:

[0028] B i = SB r (6)

[0029] Among them Obviously, as long as one reference magnetometer is specified and the misalignment error matrix S of the remaining magnetometers is obtained, the measured values of the sensors can be converted into true values, thus completing the misalignment error correction of the magnetometer array.

[0030] 2. Introduction to Firefly Optimization Algorithm

[0031] The Firefly Optimization Algorithm (FA) is a heuristic optimization algorithm based on simulating the swarm behavior characteristics of fireflies. Fireflies attract other fireflies by emitting light, and this foraging behavior is affected by light intensity and distance, which becomes the core of the Firefly Optimization Algorithm. Compared with other heuristic algorithms, the Firefly Algorithm requires fewer parameters and has good global search ability and fast convergence, and has been widely used in solving complex problems.

[0032] Next, taking the parameter setting of a single fluxgate sensor as an example, the principle and parameter selection of the Firefly Algorithm are described in detail.

[0033] The essence of the fluxgate sensor error correction is to solve 9 correction parameters, which can be regarded as being carried out in a 9-dimensional space. In this space, the position vector of each firefly represents a set of potential correction parameters, and its attractiveness reflects the quality of this set of parameters. First, initialize. Set the number of fireflies in the population as m. The position vector of each firefly is composed of 9 parameters, x i =(x i1 ; x i2 ;...; x i9 ) represents its coordinates in the 9-dimensional space. The parameters represented by each column are k x , k y , k z , b x , b y , b z, γ, α, β. The calculation of the attraction degree value is completed by where is the initial attraction degree when the distance between two fireflies is 0, ψ is the light absorption coefficient of the firefly, and r ij is the distance between the current firefly i and firefly j. During the operation of the algorithm, each firefly will move towards those fireflies with higher brightness (i.e., the attraction degree value is larger than its own). The quality of the current position of the firefly is evaluated by the fitness function. The fitness function for fluxgate sensor error correction is:

[0034]

[0035] In formula (7), B i represents the ideal magnetic field value, and N is the number of measurement postures of the fluxgate sensor. The difference between the corrected magnetic field value and the modulus of the ideal value is used as an index to evaluate the correction effect. The smaller the difference, that is, the closer the fitness function value is to 0, the better the correction effect. On the contrary, the correction effect is worse.

[0036] p i =(p i1 , p i2 ,..., p i9 ) represents the position of the i-th firefly. During the operation of the algorithm, the firefly will be attracted by brighter fireflies around it and then move towards the brighter fireflies. That is, each firefly will adjust its position according to the "best solution" around it. The moving distance is obtained by the following formula (8):

[0037]

[0038] In formula (8), t represents the number of iterations, pi represents the position of a firefly with a higher attraction degree than the j-th firefly, and r represents the distance between the i-th firefly and the j-th firefly. Among them, the initial attraction degree is set to 0.97, and the light absorption coefficient ψ is set to where ub n is the upper bound of the value of the n-th parameter, and lb n is the lower bound of the value of the n-th parameter. r ij is the distance between the current firefly i and firefly j. During the iteration process, the attraction degree is updated according to . The movement of the firefly is not only guided by the attraction force but also contains a certain degree of randomness. rand is a random perturbation, and δ is the step size factor of the perturbation.

[0039] In the firefly algorithm, each firefly first calculates the moving distance to all other fireflies. Subsequently, they evaluate the fitness value of the new position after flying towards other individuals with higher brightness than themselves. If the fitness of the new position is better than the current position, the firefly chooses to fly to the new position; otherwise, it stays at the original position. The algorithm continues until the preset maximum number of iterations is reached. At this time, the optimal firefly position searched is output as the final solution; if the maximum number of iterations is not reached, the algorithm continues to loop. In this way, the firefly population gradually gathers at the brightest point, which is the optimal solution to the problem.

[0040] 3. According to the error correction model, collect calibration data and complete the solution of error parameters

[0041] For the self-error of the fluxgate sensor, place the sensor in a uniform magnetic field environment, rotate it around the x, y, and z axes respectively, and collect 100 groups of data; use the above error correction model and firefly optimization algorithm to calculate the error parameters.

[0042] For the misalignment error of the cross-shaped magnetometer array, place the array in a uniform magnetic field environment, rotate it around the axis, and collect data. Select any one magnetometer as a reference, and use the error correction model and optimization algorithm to sequentially calculate the error parameters of the remaining magnetometers.

[0043] Next, the technical effects of the solution of the present invention will be described through simulation experiments.

[0044] In this simulation experiment, MATLAB tools were used to generate the output data of 200 fluxgate sensors in different postures. The selection method of the posture is as follows: the polar angle range is set from 0 to Π, and the azimuth angle range is set from 0 to 2Π, and random rotation is performed within this range. According to the principle of short-term invariability of the geomagnetic field, it is assumed that the sensor is in a uniform and stable geomagnetic environment, and the total field intensity is set to 51600 nT. Based on this environmental condition, by modeling and analyzing the nine error parameters of the sensor, as shown in the "preset parameters" in Table 1, the difference between its actual output value and the preset value is simulated.

[0045] (1) In order to simulate the real measurement environment, Gaussian white noise with a variance of 0.1 nT was introduced on the three measurement axes of the fluxgate sensor during the simulation process, and the initial measurement values are as Figure 3 shown. Analyze based on the measurement data of 200 different postures. The analysis results show that in the geomagnetic field environment with a modulus of 51600 nT, without correction, the measurement error of the fluxgate sensor is significant. Taking sensor 1 as an example, its error range is -1937.5009 to 1512.1245 nT. These data results show that in order to ensure the accuracy and reliability of the measurement data, the original measurement data must be corrected.

[0046] The particle swarm optimization algorithm and the firefly algorithm are respectively used to solve the calibration parameters. The parameters of the FA are set as follows: the population size of fireflies is 50, the number of iterations is 500, the step size factor is 1.5, the step size attenuation coefficient is 0.85, and the initial attractiveness is set to 0.88. The PSO also sets the population size to 50, the number of iterations to 500, both learning factors to 1.5, and the inertia weight to 1. The fitness function used by both algorithms is the one given in the previous text.

[0047] The simulation results are shown in Table 1. Taking Sensor 1 as an example, it can be seen that the FA and PSO algorithms are basically the same in the solution accuracy of the sensitivity coefficient and non-orthogonality, and they can both accurately calculate these six parameters. However, the PSO algorithm has a relatively large deviation in the solution of the zero bias, and the maximum error of Sensor 1 can reach 41.10%. The FA algorithm has a relatively small deviation in the solution of the zero bias, and the estimated error of Sensor 1 is less than 1.67%.

[0048] Table 1 Estimation table of error parameters for different calibration methods

[0049]

[0050] As Figure 4 shown, it is the total field value measurement error before and after calibration. Before calibration, the error range of Sensor 1 is -1937.5009 to 1512.1245 nT. After calibration, the error ranges corrected by the PSO algorithm are -56.1975 to 42.1775 nT in turn; the error range after calibration by the FA algorithm is between -14.7669 and 8.2873 nT.

[0051] (2) Array misalignment error

[0052] (2.1) Comparison of different optimization algorithms

[0053] Table 2 Estimation table of error parameters for different calibration methods

[0054]

[0055] Note: Sen.2 - 4 in the table represent Sensors 2 - 4 in turn.

[0056] During the empty mining, the measured values of the four sensors should be exactly the same. Therefore, taking Sensor 1 as the reference, the other sensors are calibrated to solve the misalignment error. The particle swarm optimization algorithm and the firefly algorithm are respectively used to solve the misalignment error parameters. The parameters of the FA are set as follows: the population size of fireflies is 50, the number of iterations is 200, the step size factor is 1.5, the step size attenuation coefficient is 0.85, and the initial attractiveness is set to 0.88. The fitness function is Among them, j represents the number of postures, N represents the total number of postures which is 200; i represents the i-th sensor. The different measurement values of sensors 2, 3, and 4 after single-sensor calibration are respectively brought into the FA algorithm, and the misalignment errors of the three sensors are obtained in sequence. As shown in Table 2, it can be seen that the capabilities of the PSO algorithm and the FA algorithm are comparable when solving the misalignment error. When solving the misalignment error parameters of the sensors, the deviation of γ of the three sensors is the largest, which are 1.47%, 1.79%, and 1.38% in sequence.

[0057] As Figure 5 shown, they are the three-axis measurement errors of each sensor before and after misalignment error correction. After correction, the maximum error values of sensor 2 are reduced to 0.84%, 2.82%, and 0.74% of those before correction in sequence; those of sensor 3 are reduced to 0.77%, 3.46%, and 0.85% of the error values before correction in sequence; those of sensor 4 are reduced to 1.58%, 3.69%, and 0.76% of the error values before correction in sequence.

[0058] (2.2) Comparison of different calibration strategies

[0059] According to Method 1, the parameters of FA are set as follows: the population number of fireflies is 50, the number of iterations is 200, the step factor is 1.5, the step decay coefficient is 0.85, and the initial attractiveness is set as 0.88. The fitness function is where j represents the number of postures, N represents the total number of postures which is 200, and E = G real - G idea , G real represents the measured value of the magnetic gradient tensor matrix, and G idea represents the theoretical value of the magnetic gradient tensor matrix.

[0060] Table 3 Error parameter estimation table of different calibration strategies

[0061]

[0062] The simulation results are shown in Table 3. It can be seen that the error parameters obtained by Method 1 are quite different from the preset values, and the error parameters obtained by Method 2 are very close to the preset values. As Figure 6 shown, it is the comparison chart of the sensor measurement errors before and after calibration of different calibration strategies. Obviously, when taking sensor 1 as the reference, Method 1 cannot achieve calibration for sensor 2 and sensor 4.

[0063] This is because the values used in the tensor matrix are relatively limited, and it can be clearly seen from the calculation formula of tensor G that if sensor 1 is used as the reference, sensors 2 and 4 have no effective connection with the reference sensor. When calculating the tensor matrix, the relative relationships between sensor 1 and sensor 3, and between sensor 2 and sensor 4 are mainly considered. Therefore, when using sensor 1 as the reference, method one can only accurately correct sensor 3, and cannot correct sensors 2 and 4. In method two, all the data measured by the sensors are utilized, and it is basically possible to achieve the purpose of aligning the coordinates of sensors 2, 3, and 4 with the reference sensor 1.

[0064] Compared with the related technologies, the error correction method for the cross-shaped magnetic gradient tensor measurement system provided by the present invention has the following beneficial effects:

[0065] 1. High-precision correction: In the zero-bias parameter estimation, the biases of the firefly optimization algorithm (FA) in both the bias and the misalignment angle error bias are relatively small (less than 2%), which is significantly better than the 41% bias of the particle swarm optimization algorithm (PSO).

[0066] 2. Low-cost implementation: By adopting the scalar correction method, only a high-precision magnetometer is required to monitor the background magnetic field, without the need for a magnetic shielding barrel or a non-magnetic turntable, reducing the equipment cost.

[0067] 3. Real-time and on-site applicability: Through the rapid convergence of the optimization algorithm iteration, the error correction can be completed in real time in the field environment, improving the flexibility of system deployment.

[0068] 4. Comprehensive error coverage: Coupling the single-sensor error and the array misalignment error model to solve the problem of the decrease in measurement accuracy caused by multiple error sources.

[0069] 5. Strong robustness: The global search ability of the FA algorithm avoids local optima and meets the solution requirements of complex non-linear error parameters. Description of the Drawings

[0070] Figure 1 It is a non-orthogonal error model diagram of the triaxial fluxgate sensor in the present invention;

[0071] Figure 2 It is a schematic diagram of the magnetometer array: (a) cross-shaped magnetometer array, (b) schematic diagram of the coordinate system conversion of the sensor misalignment error;

[0072] Figure 3 It is a schematic diagram of the initial magnetic measurement value of the sensor in the present invention;

[0073] Figure 4 It is a comparison diagram of the total magnetic field error value before and after correction in the present invention;

[0074] Figure 5 It is a comparison diagram of the sensor measurement error before and after correction in the present invention;

[0075] Figure 6 This is the comparison chart of the measurement errors of the sensors before and after calibration in the present invention;

[0076] Figure 7 This is the flowchart of the error correction method for a cross-shaped magnetic gradient tensor measurement system proposed by the present invention;

[0077] Figure 8 This is the comparison chart of the total magnetic field error value before and after calibration in the present invention;

[0078] Figure 9 This is the schematic diagram of the error values before and after the misalignment error correction of the magnetic gradient tensor positioning system in the present invention. Specific embodiments

[0079] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0080] The present invention proposes an error correction method for a cross-shaped magnetic gradient tensor measurement system. The specific implementation process of this correction is as follows.

[0081] The entire magnetic gradient tensor positioning system is built from 4 fluxgate sensors to be calibrated, a self-made non-magnetic cross-shaped bracket (the main material of the bracket is PC material, mainly fixed by gluing method, and reinforced with non-magnetic brass parts), a data acquisition card, a host computer and an outdoor power supply, and a certain high-precision fluxgate sensor is used as the reference sensor.

[0082] First, conduct tests on individual fluxgate sensors: Fix the sensor to be calibrated and the reference sensor together with tape, place them on the bracket, and rotate them to different orientations around the x, y, and z axes respectively, and collect 24 groups of data in sequence. The four sensors to be calibrated are carried out in sequence, and a total of 72 groups of data are collected for each sensor.

[0083] In an environment with basically no external magnetic interference, the output of the total magnetic field intensity should be close to the background geomagnetic field intensity. As Figure 8 shown, the differences between the measured values of the total magnetic field of the four sensors to be measured before and after calibration and the measured value of the total magnetic field of the reference sensor are shown in sequence, that is, the total magnetic field error values of the four sensors before and after calibration. It can be seen from this figure that after calibration, the absolute value of the maximum error of sensor 1 is reduced from 411.99 nT to 21.85 nT, which is reduced to 5.30% of that before calibration; the absolute value of the maximum error of sensor 2 is reduced to 4.48% of that before calibration; the absolute value of the maximum error of sensor 3 is reduced to 9.98% of that before calibration; the absolute value of the maximum error of sensor 4 is reduced to 11.12% of that before calibration.

[0084] Secondly, a magnetic gradient tensor array experiment is carried out: the sensor to be calibrated is fixed on the bracket, and the bracket is rotated around the z-axis to collect 32 groups of measurement data in different orientations. The misalignment error of the cross-shaped magnetic gradient tensor measurement system is corrected by using the experimental data.

[0085] As Figure 9 , it is a schematic diagram of the error values before and after the misalignment error correction of the magnetic gradient tensor positioning system. It can be seen that the absolute value of the maximum measurement error of the x-axis of sensor 2 is reduced to 16.89% of the original, the absolute value of the maximum measurement error of the y-axis is reduced to 15.80% of the original, and the absolute value of the maximum measurement error of the z-axis is reduced to 14.59% of the original; the absolute value of the maximum measurement error of the x-axis of sensor 3 is reduced to 5.60% of the original, the absolute value of the maximum measurement error of the y-axis is reduced to 10.11% of the original, and the absolute value of the maximum measurement error of the z-axis is reduced to 8.86% of the original; the absolute value of the maximum measurement error of the x-axis of sensor 4 is reduced to 13.21% of the original, the absolute value of the maximum measurement error of the y-axis is reduced to 5.37% of the original, and the absolute value of the maximum measurement error of the z-axis is reduced to 12.02% of the original.

[0086] The above are only the embodiments of the present invention, and do not limit the patent scope of the present invention accordingly. Any equivalent structure or equivalent process transformation made by using the content of the specification and drawings of the present invention, or directly or indirectly applied in other related technical fields, shall be equally included in the patent protection scope of the present invention.

Claims

1. An error correction method for a cross-shaped magnetic gradient tensor measurement system, characterized in that, It includes the following steps: Step 1: Establish an error model for the cross-shaped magnetic gradient tensor measurement system. The error model includes the three-axis sensitivity error, non-orthogonal error, zero-bias error of a single fluxgate sensor, and the coupling relationship of the misalignment errors of multiple sensors in the magnetometer array; Step 2: According to the error correction model, collect calibration data, construct an optimization objective function based on the firefly optimization algorithm (FA), and complete the solution of error parameters; The objective function: For a single fluxgate sensor, the sum of the squared residuals between the corrected magnetic field value and the theoretical value is used as the core index, and a sensitivity weight matrix, a non-orthogonal correction coefficient, and a zero-bias matrix are introduced; for the misalignment error of the magnetometer array, with a reference sensor as the benchmark, a fitness function is designed based on the magnetic field difference between the remaining sensors and the reference sensor, and a misalignment error rotation matrix is introduced; Step 3: Substitute the optimized parameters into the error model, compensate the original measurement data in real time, and output the corrected magnetic gradient tensor value.

2. The error correction method of the cross-shaped magnetic gradient tensor measurement system according to claim 1, characterized in that The final error correction model of the single fluxgate sensor in Step 1 is: B i = T -1 (B m - b) Among them, B m is the measured value of the fluxgate sensor, and B i is the actual magnetic field value, b = [b x , b y , b z T is the zero bias error, T -1 is the inverse matrix of T, and T is the comprehensive error matrix.​ 3. The error correction method of the cross-shaped magnetic gradient tensor measurement system according to claim 1, characterized in that The misalignment error model of the magnetometer array in Step 1 is: B i = SB r Among them, B r represents the measured value containing misalignment errors, and B i represents the ideal value. are the rotation matrices about the X-axis, Y-axis, and Z-axis respectively, ψ is the roll angle, is the pitch angle, and θ is the azimuth angle.

4. The error correction method of the cross-shaped magnetic gradient tensor measurement system according to claim 1, characterized in that The firefly optimization algorithm is a heuristic optimization algorithm based on the group behavior of fireflies, which simulates the foraging behavior of fireflies attracting companions by emitting light. Its core mechanism is: the brightness of a firefly is related to its fitness value, and individuals with higher brightness attract individuals with lower brightness to move towards them, thus achieving global optimization; The calculation formula for the attractiveness value of a firefly is: Among them, is the initial attractiveness when the distance between two fireflies is 0, ψ is the light absorption coefficient of the firefly, and r ij is the distance between the current firefly i and firefly j; During the operation of the algorithm, each firefly will move towards those fireflies with higher brightness, that is, towards fireflies with a larger attractiveness value than itself. The quality of the current position of the firefly is evaluated by the fitness function. The fitness function for the error correction of the fluxgate sensor is: Among them, B i represents the ideal magnetic field value, N is the number of measurement postures of the fluxgate sensor, and the difference between the corrected magnetic field value and the modulus of the ideal value is used as an index to evaluate the correction effect. The smaller the difference, that is, the closer the fitness function value is to 0, the better the correction effect, and vice versa, the worse the correction effect; p i = (p i1 , p i2 ,..., p i9 ) represents the position of the i-th firefly. During the running of the algorithm, the firefly will be attracted by brighter fireflies around it and then move towards the brighter fireflies. That is, each firefly will adjust its position according to the "optimal solution" around it. The calculation formula for its moving distance is as follows: where t represents the number of iterations, p i represents the position of a firefly with a higher attractiveness than the j-th firefly, and r represents the distance between the i-th firefly and the j-th firefly; among them, the initial attractiveness is set to 0.97, and the light absorption coefficient ψ is set to where ub n is the upper bound of the value of the n-th parameter, and lb n is the lower bound of the value of the n-th parameter, r ij is the distance between the current firefly i and firefly j, and the attractiveness is updated according to during the iteration process; the movement of the firefly is not only guided by the attraction but also contains a certain degree of randomness. rand is a random perturbation, and δ is the step size factor of the perturbation.

5. The error correction method of the cross-shaped magnetic gradient tensor measurement system according to claim 1, characterized in that, The calibration data collection method in Step 2 is: For the self-error of the fluxgate sensor, place the sensor in a uniform magnetic field environment, rotate it around the x, y, and z axes respectively, collect 100 groups of data, and use the error correction model and the firefly optimization algorithm to calculate the error parameters; For the misalignment error of the cross-shaped magnetometer array, place the magnetometer array in a uniform magnetic field environment, rotate it around the axis, collect data, select any one magnetometer as the reference, and use the error correction model and the optimization algorithm to sequentially calculate the error parameters of the remaining magnetometers.

6. The error correction method for the cross-shaped magnetic gradient tensor measurement system according to claim 1, characterized in that, The method also includes a verification step for the correction effect: Before and after correction, calculate the total field error of the sensor measurement values respectively, compare the correction accuracies of the particle swarm optimization algorithm and the firefly optimization algorithm, and verify the advantages of the firefly optimization algorithm in zero-bias parameter estimation and misalignment error correction.

Images